We elaborate on the proposal that the observed acceleration of the Universe
is the result of the backreaction of cosmological perturbations, rather than
the effect of a negative-pressure dark-energy fluid or a modification of
general relativity. Through the effective Friedmann equations describing an
inhomogeneous Universe after smoothing, we demonstrate that acceleration in our
local Hubble patch is possible even if fluid elements do not individually
undergo accelerated expansion. This invalidates the no-go theorem that there
can be no acceleration in our local Hubble patch if the Universe only contains
irrotational dust. We then study perturbatively the time behavior of
general-relativistic cosmological perturbations, applying, where possible, the
renormalization group to regularize the dynamics. We show that an instability
occurs in the perturbative expansion involving sub-Hubble modes, which
indicates that acceleration in our Hubble patch may originate from the
backreaction of cosmological perturbations on observable scales.

Now they seem to claim that most arguments presented agains their model are wrong. Could someone more knowledged enlighten us on these issues?

I've talked to various people and a majority seems to think that doing stuff in Fourier space is wrong as you easily break causality and causality is the biggest problem here: so, conceptually, what is their main argument why naive causal counter-arguments are wrong?

In fact, in the new paper Kolb et al acknowledge that their model was wrong and that superhorizon perturbations in a (vorticity-free) dust universe cannot lead to acceleration.

They also discuss various issues related to subhorizon perturbations, with the bottom line that after density perturbations go non-linear, linear theory breaks down and non-linear methods are needed to evaluate backreaction effects.

I don't quite understand what you mean about Fourier space and causality.

The patch of space which has been in causal contact is (in inflationary settings) much bigger than the apparent horizon. Hui and Seljak argued that the effect of this is restricted to setting the initial conditions non-locally. They then argued that regardless of the initial conditions, the expansion cannot accelerate if only superhorizon perturbations are present (in the case of rotationless dust). The validity of this conclusion seems to be now generally accepted. (Note that this does not mean that superhorizon perturbations cannot have a local effect in other situations, e.g. in scalar field driven inflation.)

Woodard and collaborators have studied the effects of superhorizon perturbations in the context of quantized gravity and QFT in curved spacetime for some time, and their explicit computations confim that at least in those settings superhorizon perturbations can have a physical effect which is not resricted to setting the parameters of a FRW background; see e.g. gr-qc/0408080 .

"In Ref. [5] a deviation from the pure matter-dominated evolution was obtained by a combination of suband super-Hubble modes generated by inflation, the latter being improperly used to amplify the backreaction."

In fact, I think this is as close as they get to retracting their original claim.
The rest of the paper pretends that there is a deeper truth to their original claim, although what they come up with (e.g., Eq. 78), is completely different: It is UV divergent not IR, and its net value is supposed to become significant and not its variance.

Also, the term that they end up suggesting as replacement for dark energy is just one in the gradient expansion (with six gradients). They also acknowledge that there is no reason for this term to dominate over others. I suspect that observations of large scale structure or at galactic scales can rule out such severe deviations from Newtonian gravity, as such corrections (with so many gradients) should be very clumpy.

To summarize:

1- The original claim is out.
2- The new suggestion is still highly speculative, and probably wrong, as it is severely constrained by the required smoothness of dark energy.

"In fact, in the new paper Kolb et al acknowledge that their model was wrong and that superhorizon perturbations in a (vorticity-free) dust universe cannot lead to acceleration. "

I am sorry, where they write this ?

Page 6, end of the second full paragraph:

"In this paper, we will show that the deviation from a matter-dominated background is entirely due to the nonlinear evolution of sub-Hubble modes which may cause a large backreaction (technically due to the disappearance of the filter modeling the volume average), while the super-Hubble modes play no dynamical role."

Page 14, end of the first paragraph of section A (which is dedicated to deriving this result):

"the effect of pure super- Hubble perturbations is limited to generating a true local curvature term which may be important but can not accelerate the expansion of the Universe."

Page 19, beginning of the last paragraph of section A:

"if only long-wavelength perturbations were present, at large times the line-element would take the form of a curvature dominated Universe, with , where Cij(x) is a function of spatial coordinates only."

Page 27, beginning of the third paragraph of the section Conclusions:

"Through the renormalization group technique, we have then shown that super-Hubble modes can be resummed at any order in perturbation theory yielding a local curvature term $\sim a_D^{-2}" at large times."

Also, the term that they end up suggesting as replacement for dark energy is just one in the gradient expansion (with six gradients). They also acknowledge that there is no reason for this term to dominate over others. I suspect that observations of large scale structure or at galactic scales can rule out such severe deviations from Newtonian gravity, as such corrections (with so many gradients) should be very clumpy.

The terms with the highest number of derivatives (at each order in Φ) are the Newtonian ones. The question is whether terms with a smaller number of derivatives could important. (Though, as Kolb et al correctly note, a non-perturbative treatment would be needed to discuss the issue, so talking about orders of perturbation theory is probably not very meaningful.)

Of course, dark energy was introduced precisely because deviations from the FRW dust behaviour (which agrees with the Newtonian average result, for closed spatial sections; see astro-ph/9510056 ) are seen on large scales.

For a good new overview of the issues, I recommend gr-qc/0506106 , "The universe seen at different scales" by Elis and Buchert.

Niayesh Afshordi wrote:

2- The new suggestion is still highly speculative, and probably wrong, as it is severely constrained by the required smoothness of dark energy.

The terms with the highest number of derivatives (at each order in Φ) are the Newtonian ones. The question is whether terms with a smaller number of derivatives could important. (Though, as Kolb et al correctly note, a non-perturbative treatment would be needed to discuss the issue, so talking about orders of perturbation theory is probably not very meaningful.)

Newtonian terms (at any order) cannot lead to any back-reaction other than a boundary term. That's why they are hoping post-Newtonian terms can do the trick.

Syksy Rasanen wrote:

Of course, dark energy was introduced precisely because deviations from the FRW dust behaviour (which agrees with the Newtonian average result, for closed spatial sections; see astro-ph/9510056 ) are seen on large scales.

I don't think this statement is correct. Newtonian and GR dymaincs (and back-reaction) are different through post-Newtonian terms. QD (as defined by Buchert et al.) is a total derivative in a flat space, but not necessarily in a curved one.

Syksy Rasanen wrote:

Niayesh Afshordi wrote:

2- The new suggestion is still highly speculative, and probably wrong, as it is severely constrained by the required smoothness of dark energy.

Could you elaborate on the constraints from smoothness?

Dark Energy is supposed to be relatively smooth (at least observationally) as it only affects the cosmic dynamics through changing the background evolution. On the other hand, a term like δ2v2 is probably very different within a galaxy cluster and within a void.

I don't think this statement is correct. Newtonian and GR dymaincs (and back-reaction) are different through post-Newtonian terms. QD (as defined by Buchert et al.) is a total derivative in a flat space, but not necessarily in a curved one

I don't see a contradiction between my statement and yours. Because the Newtonian backreaction term is a total derivative, it does not lead to any backreaction for closed spatial sections. Therefore the Newtonian average equations agree with the (spatially flat) FRW dust equations.

Niayesh Afshordi wrote:

Dark Energy is supposed to be relatively smooth (at least observationally) as it only affects the cosmic dynamics through changing the background evolution. On the other hand, a term like δ2v2 is probably very different within a galaxy cluster and within a void.

Backreaction is the difference between the average behaviour of a given domain and the behaviour of a corresponding smooth and isotropic domain. So it doesn't make sense to talk about 'backreaction' within a single cluster. (Unless one is treating the cluster as homogeneous and isotropic in order to evaluate some average quantities, but that's presumably not what you mean?)

On the other side, dark energy effects beyond the background level are not ruled out. For example, dark energy could have perturbations, and even affect the virialisation of collapsing structures (e.g. astro-ph/0401504,astro-ph/0505308).

Hi all,
In gr-qc/0509108, Ishibashi and Wald refute the claims of backreaction being the reason for the observed local acceleration. They have an example showing that averaging could show up an artificial acceleration. Just after reading this, I saw astro-ph/0510059. This is a "white paper" on the dark matter problem by Bean, Carrol and Trodden. They feel that the possibility of explaining "dark energy" by incorporating higher-order corrections is still possible. Is this the common feeling ?? The poll here certainly does not point that way :-?

I am joining now this forum, and I would like to say my opinion on the subject.

As I stressed in my paper astro-ph/0503715 I have shown that the perturbative series (that is, assuming the density contrast δ and the gravitational potential φ be much smaller than 1) breaks down at redshifts of order 1.

This is the series of the PostNewtonian terms.

It could be that this is just an artifact of the perturbative expansion, but so far we have no serious result.
The only serious criticism made to this is that in a Newtonian and PostNewtonian expansion, the leading term "could" be of the form (for dimensional arguments) :

< φδ > (where <> is the spatial average)

and this is typically 10 - 5

Note however that this is only a guess. It is very important to have the correct form of the correction. As Niayesh says it is true that in the newtonian approach φ is small.....but I believe it is not true the conclusion that the backreaction is necessarily small.
There is an additional quantity in fact which is δ, and δ > > 1.

Suppose that the backreaction appears for example as:

Dimensional anlysis would say that this is of the same order of δφ...but it is not!
And in fact it is of order 1.

I've been trying to understand these papers, but I have some naive questions that I was hoping someone could help me with:

On p.21 of astro-ph/0506534, they repeat an argument that the Newtonian terms at order n, of the form (∂2φ)n, are total derivatives. This seems obvious to everyone who is knowledgeable about such matters, but I'm having trouble understanding why it should be the case. Such terms aren't total derivatives in the usual sense, and indeed the later argument relies on terms like (with at least four gradient operators - but at order 2n > 4 in the expansion) being arbitrarily large. I assume there is a subtlety with the averaging that I am missing?

On p. 22, they claim that diverges like lnΛ, for some ultraviolet cutoff Λ. Although the argument isn't written out, it seems to be that for a term with p + q = N gradient operators,
(up to factors of i and minus signs), where T(k) is the CDM transfer function, for large k (and 2π / keq is the comoving size of the horizon at matter-radiation equality), an overline is the ensemble average, and angle brackets are the spatial average. Then it follows that
if the integral is dominated by its upper limit, assuming a scale invariant spectrum (as they do), and is the amplitude of perturbations. Thus for N = 4 this diverges like lnΛ.

My understanding was that φ was the gravitational potential after inflation (cf. the discussion on p.15) and was the initial condition for their "renormalization" procedure. Is there an easy way to see why the CDM transfer function should enter here?

Assuming that T(k) ought to be present, the point of the lengthy calculations in Sec. III.A & III.B seems to be that for sufficiently high-order terms in the gradient expansion (), the averages can become arbitrarily large. This conclusion appears to rely on the scale-invariant spectrum extending to arbitrarily large wavenumbers. However, I would naively have expected that the spectrum is really cut off at a scale corresponding to the horizon size at the end of inflation. If this is done, then
where a subscript 0 indicates the end of inflation. Since keq is less than k0 this quantity is presumably tiny?

It's claimed (on p.24) that for large a, the so-called "renormalized" gravitational potential Ψ behaves as in Eq. (71), Eq. (75) when truncated to 2 gradients or 4 gradients (respectively), and in particular grows like (lna)n when truncated to 2n gradients. However, since they seem to assume , in (eg.) Eq. (71), the argument of the logarithm goes to zero for
Presumably this is like the Landau pole in QED, and would seem to indicate that their "renormalized" perturbation theory breaks down at a finite value of a?